Chapter 1
Perspective in Art: The Horizon
I like to draw and paint what I see. To achieve realism in a drawing, I have learned that I need
to be able to distinguish the obvious from the apparent; to distinguish what I am sure I must
be seeing - or what I might measure - from what I really do see. A common quality of
drawings by inexperienced artists (children, for instance) is that they often appear flat.
Animals such as dogs are usually drawn facing either directly left or right and the point of view
is neither from above nor below. People are also usually drawn “at eye level” and from only
one point of view, facing directly outward from the paper, having the same general shape as a
gingerbread man. Railroad tracks appear as if viewed from above. Houses are likewise drawn
from a viewpoint where only one side is visible.
Figure 1-1. A flat boy and his dog; a flat house by the railroad tracks.
Eventually the developing artist will come to realize that more than one side of the house can
be seen (and drawn) at the same time. If he is able to distinguish what he “knows” from what
he sees, he will be successful. This ability serves not only the artist but the scientist as well,
and that is why I want to emphasize it. In Galileo's time, everyone “knew” that heavier objects
must fall faster than light ones. But Galileo took a closer look and we have all learned from his
work. The beginning artist intuitively “knows” that all the corners of his house are equal in
height and may be inclined to draw them that way. He can get a tape measure and prove it.
But if he looks closely and draws what he sees rather than what he knows, the results will be
different.
4
Figure 1-2. The same scene in perspective.
There are very well-established rules for the use of perspective in Realistic art. Note that I
used (and capitalized) the word “Realistic.” This is not a value judgment, but rather a
classification of art. Realism in art is characterized by how much it corresponds to the viewer's
experience. Cubism, in contrast, makes a deliberate attempt to represent multiple points of
view in the same image (Figure 1-3). It does not look “realistic” because this is not the point.
Cubism's greatest prevalence was in the early 20th century and in the context of the
popularization of photography. With photography encroaching on its domain, art diversified.
My understanding of art history is superficial at best, but without drawing any conclusions I
would like to point out a coincidence in contemporary science. Cubism's time of popularity
was also shortly after Einstein published his theory of special relativity, from which we learned
that the time order of two events could depend on one's perspective. Rather than speaking of
time as separate from three-dimensional space, scientists were increasingly speaking of a
four-dimensional “spacetime.” This “extra dimension” had seized the popular imagination as
early as H.G. Wells' The Time Machine and the idea took an even greater hold after Einstein's
theories were published. It seems no great surprise, therefore, that the “extra dimension”
began to appear in art around this time. Why might it have been called “Cubism”? A drawing
or painting is flat and squarish. What happens when you add another dimension of depth to a
square? It becomes a cube.
5
Figure 1-3. A cubist portrait sketch.
In contrast to Cubist art, Realistic drawings and paintings have one and only one point of
view, and there are rules on how to accurately represent this point of view. I will simplify them
here. These rules are most readily demonstrated in the case of parallel lines and right angles;
for instance, when depicting buildings, fences, railroad tracks, and so on. In the simple cases
taught to beginning art students, the point of view is horizontal (neither up nor down, but
parallel to the ground) and the horizon is visible somewhere in the picture, forming a line from
left to right and theoretically extending beyond the left- and right-hand edges of the paper or
canvas. For the sake of generality, this horizon is called the “eye level line,” but since it's
understood that we're discussing horizontal points of view for simplicity's sake, I'll simply call it
the “horizon.” In Figure 1-2, the horizon line passes behind the house.
The main principle at work in perspective drawing is very simple: Things appear smaller as
they get further away. You see this all the time. Look at a group of far-away people, and the
distance between them looks very small. Their heads appear closer to their feet. You can hold
up your thumb and forefinger in front of them and imagine that they're only a couple of inches
tall. In Figure 1-2, the dog occupies less space on the page than the boy's face; it appears
smaller, but we rationalize this by supposing that the dog is in the background behind the boy
6
rather than floating above his hand.
Another way that we see this principle at work is when things appear to change shape and
size depending on the angle of our viewpoint. Long things look shorter when one end is
nearer than the other. The more distant end appears smaller than the nearer one, making the
entire object appear both shorter and narrowing toward one end (as seen in the rectangular
block in Figure 1-4). The floor and ceiling and opposite walls of a hallway look closer together
the further you look down the hall (Figure 1-5).
Figure 1-4. A block viewed from two angles.
Figure 1-5. Numbered doors in a hallway.
This principle results in a couple of rules for perspective drawing:
Rule #1: A series of regularly spaced objects will appear to come closer together as the
series recedes toward the horizon.
In Figure 1-2, the distant railroad ties appear closer together than the near ones. In Figure 15, door 3 appears closer to door 2 than to door 4. The telephone poles at the side of the road
appear closer together the further down the road you look. So do the cross-ties on the railroad
track and the posts in the fence. If you were to go to the side of the road with a tape measure,
you would find that the poles do not in fact get closer together. But if you point a camera down
the road, take a picture, print it, and then use a ruler to measure the apparent distance
between poles, that distance will change as you move across the picture.
Rule #2: Horizontal lines, if they point in the same compass direction, diverge from a common
point on the horizon called their “vanishing point.”
7
Figure 1-6.
The lines on a long, straight section of highway seem to meet in the far-off distance. They are
joined by the power and telephone lines hanging from the roadside poles (Figure 1-6). The
location of a line's vanishing point is independent of how high this horizontal line is (namely,
whether it's at the level of the road or the power lines); it only depends on which compass
direction the line takes. Let's imagine a northward point of view and a line on the ground
directly in front of the viewer. If the line is pointing directly toward and away from the viewer
(passing under his feet), then it must be pointing north/south, and the vanishing point for that
line is in the center of the horizon line (Figure 1-7). Turn the line clockwise and the line's
vanishing point moves to the right (Figure 1-8). Turn it counterclockwise and it moves left.
Turn it further and the vanishing point moves “off the page.” The further the lines turns, the
further away the vanishing point is from the edge of the page.
Figure 1-7.
8
Figure 1-8.
Figure 1-9.
Turn the line far enough, so that it is pointing east-west, and the theoretical vanishing point is
now an infinite distance from the page (Figure 1-9). Is the vanishing point on the left side or
right side? Either. Both. One of the interesting things about infinity is that – being the
impossible number - it doesn't obey the rules that hold for real numbers. We will encounter
the idea of infinity many times in our exploration of relativity.
Let's start a drawing with two lines which vanish into a common point (Figure 1-10).
Figure 1-10. Parallel lines.
Look at it for a moment and think what these two lines might depict. Whether it becomes a
vertical or horizontal object depends entirely on how you connect the lines. I can turn it into a
fence (Figure 1-11) or railroad tracks (Figure 1-12). What did it look like to you?
9
Figure 1-11. Railroad tracks.
Figure 1-12. Fence.
You'll notice that the railroad ties and fence posts are all drawn parallel to one another. The
railroad ties all lie horizontally, so the rule says that they should meet at a common vanishing
point. In this case, that vanishing point is a theoretical one an infinite distance off either side
of the page. But what about the vertical fence posts? What do the rules say about them? I'm
going to introduce a temporary rule here which we will throw out in a later chapter.
Rule #3: Vertical lines are parallel.
We have several assumptions at work here. One, already stated, is that the point of view is
horizontal, or parallel to the ground. The second one, which we come to now, is that the
image on the paper represents a small angle of the viewpoint. In other words, we're only
drawing what's directly in the viewer's line of sight, not what is in their peripheral vision. I will
explain why we make this assumption. A fisheye lens is a lens that focuses images from a
very wide angle and it creates an interesting distortion of the image. It is that angular width of
the viewpoint that is the central issue in perspective drawing. The reason things appear
smaller as they get further away is that they fit in a smaller angle of our total view.
In chapter eight, we'll throw out rule #3 along with our two assumptions and then we will see
what happens. Your assignment now, and at the end of each chapter, will be to spend some
time pondering and observing. It takes time to fully absorb some concepts, and the concept of
perspective requires as much time as a Biblical parable or a Zen koan. Go outside and look at
the railroad tracks and telephone poles. Spend some time analyzing what you really see. If
you really want to understand, make some drawings. Then come back and read the chapter
again. Make a habit of spending time with each of the concepts in this book, because you will
find it too heavy a meal to eat all in one sitting.
10
Figure 1-13. Self-portrait by M. C. Escher. The reflection in the globe is similar to an image seen through an
extremely wide-angle lens.
11